ThmDex – An index of mathematical definitions, results, and conjectures.
Result R3261 on D207: Bernoulli random boolean number
Probability for two independent Bernoulli random numbers to coincide
Formulation 1
Let $X \in \text{Bernoulli}(\theta_X)$ and $Y \in \text{Bernoulli}(\theta_Y)$ each be a D207: Bernoulli random boolean number such that
(i) $X, Y$ is an D2713: Independent random collection
Then
(1) \begin{equation} \mathbb{P}(X = Y) = 1 - \theta_X - \theta_Y + 2 \theta_X \theta_Y \end{equation}
(2) \begin{equation} \mathbb{P}(X \neq Y) = \theta_X + \theta_Y - 2 \theta_X \theta_Y \end{equation}